mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
u v : E
u_in : (u, v).1 ∈ B ∩ s
v_in : (u, v).2 ∈ B ∩ s
conv : Convex ℝ (B ∩ s)
diff : DifferentiableOn ℝ f (B ∩ s)
⊢ ‖f (u, v).2 - f (u, v).1 - (f' (u, v).2 - f' (u, v).1)‖ ≤ ε * ‖(u, v).2 - (u, v).1‖

Tactic:
have bound : ∀ z ∈ B ∩ s, ‖fderivWithin ℝ f (B ∩ s) z - f'‖ ≤ ε := by
      intro z z_in
      have h := hδ z
      have : fderivWithin ℝ f (B ∩ s) z = fderiv ℝ f z := by
        have op : IsOpen (B ∩ s) := isOpen_ball.inter s_open
        rw [DifferentiableAt.fderivWithin _ (op.uniqueDiffOn z z_in)]
        exact (diff z z_in).differentiableAt (IsOpen.mem_nhds op z_in)
      rw [← this] at h
      exact le_of_lt (h z_in.2 z_in.1)

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
u v : E
u_in : (u, v).1 ∈ B ∩ s
v_in : (u, v).2 ∈ B ∩ s
conv : Convex ℝ (B ∩ s)
diff : DifferentiableOn ℝ f (B ∩ s)
bound : ∀ z ∈ B ∩ s, ‖fderivWithin ℝ f (B ∩ s) z - f'‖ ≤ ε
⊢ ‖f (u, v).2 - f (u, v).1 - (f' (u, v).2 - f' (u, v).1)‖ ≤ ε * ‖(u, v).2 - (u, v).1‖
Module: Mathlib.Analysis.Calculus.Deriv.Prod

Goal state before:
pp: case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_1
E' : ι → Type u_2
inst✝¹ : (i : ι) → NormedAddCommGroup (E' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E' i)
φ : 𝕜 → (i : ι) → E' i
h : ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x ↦ φ x i) s x
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ derivWithin φ s x = fun i ↦ derivWithin (fun x ↦ φ x i) s x

Tactic:
derivWithin,

Goal state after:
case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
x : 𝕜
s : Set 𝕜
ι : Type u_1
E' : ι → Type u_2
inst✝¹ : (i : ι) → NormedAddCommGroup (E' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E' i)
φ : 𝕜 → (i : ι) → E' i
h : ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x ↦ φ x i) s x
hsx : UniqueDiffWithinAt 𝕜 s x
⊢ (fderivWithin 𝕜 φ s x) 1 = fun i ↦ derivWithin (fun x ↦ φ x i) s x
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ U, (U \ Subtype.val '' {u | analyticOrderAt f ↑u = 0 ∨ analyticOrderAt f ↑u = ⊤})ᶜ ∈ 𝓝[≠] x

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
x : 𝕜
⊢ x ∈ U → (U \ Subtype.val '' {u | analyticOrderAt f ↑u = 0 ∨ analyticOrderAt f ↑u = ⊤})ᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
⊢ ∀ x ∈ U, (U \ {u | analyticOrderAt f u = 0 ∨ analyticOrderAt f u = ⊤})ᶜ ∈ 𝓝[≠] x

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
U : Set 𝕜
f : 𝕜 → E
hf : AnalyticOnNhd 𝕜 f U
x : 𝕜
⊢ x ∈ U → (U \ {u | analyticOrderAt f u = 0 ∨ analyticOrderAt f u = ⊤})ᶜ ∈ 𝓝[≠] x
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₁ : f =ᶠ[codiscreteWithin U] g
h₂ : EqOn f g Uᶜ
⊢ MeromorphicOn g U

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₁ : f =ᶠ[codiscreteWithin U] g
h₂ : EqOn f g Uᶜ
x : 𝕜
⊢ x ∈ U → MeromorphicAt g x
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₁ : f =ᶠ[codiscreteWithin U] g
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
⊢ MeromorphicOn g U

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
h₁ : f =ᶠ[codiscreteWithin U] g
h₂ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
x : 𝕜
⊢ x ∈ U → MeromorphicAt g x
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
⊢ ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \ {x | AnalyticAt 𝕜 f x}))

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
x : 𝕜
⊢ x ∈ U → Disjoint (𝓝[≠] x) (𝓟 (U \ {x | AnalyticAt 𝕜 f x}))
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
x : 𝕜
hx : x ∈ U
⊢ {x | (fun y ↦ AnalyticAt 𝕜 f y) x} ⊆ (U \ {x | AnalyticAt 𝕜 f x})ᶜ

Tactic:
intro x

Goal state after:
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
U : Set 𝕜
inst✝ : CompleteSpace E
f : 𝕜 → E
h : MeromorphicOn f U
x✝ : 𝕜
hx : x✝ ∈ U
x : 𝕜
⊢ x ∈ {x | (fun y ↦ AnalyticAt 𝕜 f y) x} → x ∈ (U \ {x | AnalyticAt 𝕜 f x})ᶜ
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : BinomialRing R
r s : R
k : ℕ
h : Commute r s
⊢ ∀ x ∈ antidiagonal k,
    ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =
      k.factorial • (choose r x.1 * choose s x.2)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : BinomialRing R
r s : R
k : ℕ
h : Commute r s
x : ℕ × ℕ
⊢ x ∈ antidiagonal k →
    ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =
      k.factorial • (choose r x.1 * choose s x.2)
Module: Mathlib.Analysis.Analytic.Polynomial

Goal state before:
pp: case refine_3
𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
s : Set E
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
f : E → B
hf : AnalyticWithinAt 𝕜 f s z
p✝ : A[X]
p : ℕ
i : A
hp : AnalyticWithinAt 𝕜 (fun x ↦ (aeval (f x)) (C i * X ^ p)) s z
⊢ AnalyticWithinAt 𝕜 (fun x ↦ (aeval (f x)) (C i * X ^ (p + 1))) s z

Tactic:
convert! hp.mul hf

Goal state after:
𝕜 : Type u_1
E : Type u_2
A : Type u_3
B : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CommSemiring A
z : E
s : Set E
inst✝² : NormedRing B
inst✝¹ : NormedAlgebra 𝕜 B
inst✝ : Algebra A B
f : E → B
hf : AnalyticWithinAt 𝕜 f s z
p✝ : A[X]
p : ℕ
i : A
hp : AnalyticWithinAt 𝕜 (fun x ↦ (aeval (f x)) (C i * X ^ p)) s z
x✝ : E
⊢ (aeval (f x✝)) (C i * X ^ (p + 1)) = (aeval (f x✝)) (C i * X ^ p) * f x✝
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: z : ℂ
hz : z ≠ 0
⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x) ^ 2)
    (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + 2))⁻¹ * (↑n + 1)) 0 ‖z‖ₑ

Tactic:
simpa [add_comm 1] using one_div_sub_pow_hasFPowerSeriesOnBall_zero 1 hz

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: r : ℝ
hr : r < 1
hr' : r ≠ 0
h : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
y : ℝ := ⋯
⊢ ∀ x ∈ Set.Ioi 0, x⁻¹ • ((0 + x) ^ r - 0 ^ r) = x ^ (r - 1)

Tactic:
intro x

Goal state after:
r : ℝ
hr : r < 1
hr' : r ≠ 0
h : DifferentiableAt ℝ (fun x ↦ x ^ r) 0
y : ℝ := deriv (fun x ↦ x ^ r) 0
x : ℝ
⊢ x ∈ Set.Ioi 0 → x⁻¹ • ((0 + x) ^ r - 0 ^ r) = x ^ (r - 1)
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case convert_2
a : ℝ
H : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)
this : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1
⊢ Set.EqOn (⇑Complex.reCLM ∘ (fun x ↦ (1 + x) ^ ↑a) ∘ ⇑Complex.ofRealCLM) (fun x ↦ (1 + x) ^ a)
    (Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ))

Tactic:
intro x

Goal state after:
case convert_2
a : ℝ
H : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)
this : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1
x : ℝ
⊢ x ∈ Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ) →
    (⇑Complex.reCLM ∘ (fun x ↦ (1 + x) ^ ↑a) ∘ ⇑Complex.ofRealCLM) x = (fun x ↦ (1 + x) ^ a) x
Module: Mathlib.Analysis.Analytic.Binomial

Goal state before:
pp: case convert_2
a : ℝ
this :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)
    (FormalMultilinearSeries.restrictScalars ℝ
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))
    (Complex.ofRealCLM 0) 1
⊢ Set.EqOn (⇑Complex.reCLM ∘ (fun x ↦ 1 / (1 - x) ^ ↑a) ∘ ⇑Complex.ofRealCLM) (fun x ↦ 1 / (1 - x) ^ a)
    (Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ))

Tactic:
intro x

Goal state after:
case convert_2
a : ℝ
this :
  HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)
    (FormalMultilinearSeries.restrictScalars ℝ
      (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))
    (Complex.ofRealCLM 0) 1
x : ℝ
⊢ x ∈ Metric.eball 0 (1 / ‖Complex.ofRealCLM‖ₑ) →
    (⇑Complex.reCLM ∘ (fun x ↦ 1 / (1 - x) ^ ↑a) ∘ ⇑Complex.ofRealCLM) x = (fun x ↦ 1 / (1 - x) ^ a) x
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: f : ℝ → ℝ
x : ℝ
s : Set ℝ
a : ℝ
ha : 0 < a
hf : DifferentiableWithinAt ℝ f s x
⊢ derivWithin (fun x ↦ a ^ f x) s x = log a * derivWithin f s x * a ^ f x

Tactic:
by_cases hxs : UniqueDiffWithinAt ℝ s x
  · exact (hf.hasDerivWithinAt.const_rpow ha).derivWithin hxs
  · rw [derivWithin_zero_of_not_uniqueDiffWithinAt hxs,
      derivWithin_zero_of_not_uniqueDiffWithinAt hxs,
      mul_zero, zero_mul]

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: case h'
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
this : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f (s ∩ Metric.eball x r) x
⊢ AnalyticOn 𝕜 f (s ∩ Metric.eball x r)

Tactic:
exact h'.mono inter_subset_left

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.ERealExp

Goal state before:
pp: ⊢ StrictMono exp

Tactic:
intro x

Goal state after:
x : EReal
⊢ ∀ ⦃b : EReal⦄, x < b → x.exp < b.exp
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: ⊢ StrictMono log

Tactic:
intro x

Goal state after:
x : ℝ≥0∞
⊢ ∀ ⦃b : ℝ≥0∞⦄, x < b → x.log < b.log
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp

Goal state before:
pp: case e'_5
b : ℝ≥0∞
hb : b < 1
⊢ ⊥ = ⊤ * b.log

Tactic:
rw [EReal.top_mul_of_neg (log_lt_zero_iff.2 hb)]

Goal state after:
Module: Mathlib.MeasureTheory.Covering.VitaliFamily

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
⊢ ∀ (x : X),
    ∀ ε > 0, ∃ s ∈ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}, s ⊆ closedBall x ε

Tactic:
intro x

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
x : X
⊢ ∀ ε > 0, ∃ s ∈ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}, s ⊆ closedBall x ε
Module: Mathlib.MeasureTheory.Covering.VitaliFamily

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
⊢ ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ g x, t ⊆ closedBall x ε

Tactic:
intro x

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
m0 : MeasurableSpace X
μ : Measure X
v : VitaliFamily μ
δ : ℝ
δpos : 0 < δ
s : Set X
f : X → Set (Set X)
fset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f x, t ⊆ closedBall x ε
g : X → Set (Set X) := fun x ↦ f x ∩ v.setsAt x
x : X
⊢ x ∈ s → ∀ ε > 0, ∃ t ∈ g x, t ⊆ closedBall x ε
Module: Mathlib.Analysis.Asymptotics.SuperpolynomialDecay

Goal state before:
pp: α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
⊢ SuperpolynomialDecay l k f ↔ ∀ (z : ℤ), f =o[l] fun a ↦ k a ^ z

Tactic:
by
  refine ⟨fun h z => ?_, fun h => (superpolynomialDecay_iff_isBigO f hk).2 fun z => (h z).isBigO⟩
  have hk0 : ∀ᶠ x in l, k x ≠ 0 := hk.eventually_ne_atTop 0
  have : (fun _ : α => (1 : β)) =o[l] k :=
    isLittleO_of_tendsto' (hk0.mono fun x hkx hkx' => absurd hkx' hkx)
      (by simpa using! hk.inv_tendsto_atTop)
  have : f =o[l] fun x : α => k x * k x ^ (z - 1) := by
    simpa using! this.mul_isBigO ((superpolynomialDecay_iff_isBigO f hk).1 h <| z - 1)
  refine this.trans_isBigO <| IsBigO.of_bound' <| hk0.mono fun x hkx => le_of_eq ?_
  simp [← zpow_one_add₀ hkx]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ (x : α) in t, f x ∂μ]

Tactic:
by_cases hs₀ : μ s = 0

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hs₀ : μ s = 0
⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ (x : α) in t, f x ∂μ]

case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s t : Set α
f : α → ℝ≥0∞
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hs₀ : ¬μ s = 0
⊢ ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ (x : α) in t, f x ∂μ]
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n ↦ {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
⊢ {x | f x ≤ g x}ᶜ ⊆ ⋃ n, s n

Tactic:
intro x

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ
A : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
s : ℕ → Set α := fun n ↦ {x | g x + ↑(u n) ≤ f x ∧ g x ≤ ↑↑n} ∩ spanningSets μ n
μs : ∀ (n : ℕ), μ (s n) = 0
x : α
⊢ x ∈ {x | f x ≤ g x}ᶜ → x ∈ ⋃ n, s n
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
⊢ (sInf {d | μ ≤ d + ν}).restrict s = sInf {d | μ.restrict s ≤ d + ν.restrict s}

Tactic:
have h_nonempty : { d | μ ≤ d + ν }.Nonempty := ⟨μ, Measure.le_add_right le_rfl⟩

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
⊢ (sInf {d | μ ≤ d + ν}).restrict s = sInf {d | μ.restrict s ≤ d + ν.restrict s}
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
⊢ (μ.restrict s) (t ∩ s) ≤ ν' (t ∩ s) + (ν.restrict s) (t ∩ s)

Tactic:
apply h_ν'_in

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
⊢ μ.rnDeriv ν =ᵐ[ν] 0 ↔ μ ⟂ₘ ν

Tactic:
by
  rw [← withDensity_rnDeriv_eq_zero, withDensity_eq_zero_iff (measurable_rnDeriv _ _).aemeasurable]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
mu_ai : μ (a i) < ∞
⊢ ∫ (a : α), ‖g i a‖ * ‖f a - c‖ ∂μ ≤ ∫ (a_1 : α) in a i, K * ((μ.restrict (a i)).real univ)⁻¹ * ‖f a_1 - c‖ ∂μ

Tactic:
← div_eq_mul_inv

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
mu_ai : μ (a i) < ∞
⊢ ∫ (a : α), ‖g i a‖ * ‖f a - c‖ ∂μ ≤ ∫ (a_1 : α) in a i, K / (μ.restrict (a i)).real univ * ‖f a_1 - c‖ ∂μ
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
mu_ai : μ (a i) < ∞
⊢ ∀ x ∉ a i, ‖g i x‖ * ‖f x - c‖ = 0

Tactic:
intro x

Goal state after:
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : CompleteSpace E
ι : Type u_4
a : ι → Set α
l : Filter ι
f : α → E
c : E
g : ι → α → ℝ
K : ℝ
hf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)
f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ
hg : Tendsto (fun i ↦ ∫ (y : α), g i y ∂μ) l (𝓝 1)
g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i
g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / μ.real (a i)
g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ
I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ
this : Tendsto (fun x ↦ K * ⨍ (y : α) in a x, ‖f y - c‖ ∂μ) l (𝓝 0)
i : ι
hi : support (g i) ⊆ a i
h'i : ∀ (x : α), |g i x| ≤ K / μ.real (a i)
h''i : IntegrableOn f (a i) μ
hi_int : 0 < ∫ (y : α), g i y ∂μ
mu_ai : μ (a i) < ∞
x : α
⊢ x ∉ a i → ‖g i x‖ * ‖f x - c‖ = 0
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∃ᶠ (ε : ℝ) in 𝓝[>] 0, ∃ a ∈ t, r a = ε ∧ c a = x
⊢ ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x

Tactic:
intro x

Goal state after:
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∃ᶠ (ε : ℝ) in 𝓝[>] 0, ∃ a ∈ t, r a = ε ∧ c a = x
x : α
⊢ x ∈ s → ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x